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Properties

Label 512.2.a.b.1.1
Level $512$
Weight $2$
Character 512.1
Self dual yes
Analytic conductor $4.088$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [512,2,Mod(1,512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("512.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.08834058349\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 512.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{3} -2.00000 q^{5} +2.82843 q^{7} -1.00000 q^{9} +4.24264 q^{11} -6.00000 q^{13} +2.82843 q^{15} -4.24264 q^{19} -4.00000 q^{21} -8.48528 q^{23} -1.00000 q^{25} +5.65685 q^{27} -2.00000 q^{29} -5.65685 q^{31} -6.00000 q^{33} -5.65685 q^{35} -6.00000 q^{37} +8.48528 q^{39} +6.00000 q^{41} -4.24264 q^{43} +2.00000 q^{45} +1.00000 q^{49} +2.00000 q^{53} -8.48528 q^{55} +6.00000 q^{57} +1.41421 q^{59} -6.00000 q^{61} -2.82843 q^{63} +12.0000 q^{65} +12.7279 q^{67} +12.0000 q^{69} +8.48528 q^{71} -12.0000 q^{73} +1.41421 q^{75} +12.0000 q^{77} +5.65685 q^{79} -5.00000 q^{81} +4.24264 q^{83} +2.82843 q^{87} -12.0000 q^{89} -16.9706 q^{91} +8.00000 q^{93} +8.48528 q^{95} -8.00000 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{9} - 12 q^{13} - 8 q^{21} - 2 q^{25} - 4 q^{29} - 12 q^{33} - 12 q^{37} + 12 q^{41} + 4 q^{45} + 2 q^{49} + 4 q^{53} + 12 q^{57} - 12 q^{61} + 24 q^{65} + 24 q^{69} - 24 q^{73} + 24 q^{77}+ \cdots - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.41421 −0.816497 −0.408248 − 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 − 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.24264 1.27920 0.639602 − 0.768706i \(-0.279099\pi\)
0.639602 + 0.768706i \(0.279099\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −4.24264 −0.973329 −0.486664 − 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −8.48528 −1.76930 −0.884652 − 0.466252i \(-0.845604\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 − 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) −5.65685 −0.956183
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 8.48528 1.35873
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.24264 −0.646997 −0.323498 − 0.946229i \(-0.604859\pi\)
−0.323498 + 0.946229i \(0.604859\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −8.48528 −1.14416
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 1.41421 0.184115 0.0920575 − 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 12.7279 1.55496 0.777482 − 0.628906i \(-0.216497\pi\)
0.777482 + 0.628906i \(0.216497\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 8.48528 1.00702 0.503509 − 0.863990i \(-0.332042\pi\)
0.503509 + 0.863990i \(0.332042\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 − 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 1.41421 0.163299
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 − 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 4.24264 0.465690 0.232845 − 0.972514i \(-0.425196\pi\)
0.232845 + 0.972514i \(0.425196\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.82843 0.303239
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −16.9706 −1.77900
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 8.48528 0.870572
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −4.24264 −0.426401
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 512.2.a.b.1.1 ✓ 2
3.2 odd 2 4608.2.a.p.1.2 2
4.3 odd 2 inner 512.2.a.b.1.2 yes 2
8.3 odd 2 512.2.a.e.1.1 yes 2
8.5 even 2 512.2.a.e.1.2 yes 2
12.11 even 2 4608.2.a.p.1.1 2
16.3 odd 4 512.2.b.e.257.4 4
16.5 even 4 512.2.b.e.257.3 4
16.11 odd 4 512.2.b.e.257.1 4
16.13 even 4 512.2.b.e.257.2 4
24.5 odd 2 4608.2.a.c.1.2 2
24.11 even 2 4608.2.a.c.1.1 2
32.3 odd 8 1024.2.e.n.257.2 4
32.5 even 8 1024.2.e.n.769.1 4
32.11 odd 8 1024.2.e.n.769.2 4
32.13 even 8 1024.2.e.n.257.1 4
32.19 odd 8 1024.2.e.h.257.1 4
32.21 even 8 1024.2.e.h.769.2 4
32.27 odd 8 1024.2.e.h.769.1 4
32.29 even 8 1024.2.e.h.257.2 4
48.5 odd 4 4608.2.d.j.2305.3 4
48.11 even 4 4608.2.d.j.2305.4 4
48.29 odd 4 4608.2.d.j.2305.1 4
48.35 even 4 4608.2.d.j.2305.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.b.1.1 ✓ 2 1.1 even 1 trivial
512.2.a.b.1.2 yes 2 4.3 odd 2 inner
512.2.a.e.1.1 yes 2 8.3 odd 2
512.2.a.e.1.2 yes 2 8.5 even 2
512.2.b.e.257.1 4 16.11 odd 4
512.2.b.e.257.2 4 16.13 even 4
512.2.b.e.257.3 4 16.5 even 4
512.2.b.e.257.4 4 16.3 odd 4
1024.2.e.h.257.1 4 32.19 odd 8
1024.2.e.h.257.2 4 32.29 even 8
1024.2.e.h.769.1 4 32.27 odd 8
1024.2.e.h.769.2 4 32.21 even 8
1024.2.e.n.257.1 4 32.13 even 8
1024.2.e.n.257.2 4 32.3 odd 8
1024.2.e.n.769.1 4 32.5 even 8
1024.2.e.n.769.2 4 32.11 odd 8
4608.2.a.c.1.1 2 24.11 even 2
4608.2.a.c.1.2 2 24.5 odd 2
4608.2.a.p.1.1 2 12.11 even 2
4608.2.a.p.1.2 2 3.2 odd 2
4608.2.d.j.2305.1 4 48.29 odd 4
4608.2.d.j.2305.2 4 48.35 even 4
4608.2.d.j.2305.3 4 48.5 odd 4
4608.2.d.j.2305.4 4 48.11 even 4